We present novel results linking model predictive control (MPC) and minimax optimal control theory. Specifically, we show that, for linear discrete-time systems with constraints and disturbance inputs, the closed-loop optimal solutions of a particular class of minimax optimal control problems are a class of typical MPC policies. We also present conditions which ensure that the inverse optimal MPC policies achieve a prescribed (regional) ℓ₂-gain from the disturbance input.